Preheat oven to 350°F (175°C). Line an 8x8 inch baking pan with parchment paper, leaving an overhang on the sides.
In a double boiler or microwave-safe bowl, melt butter and chopped semisweet chocolate together until smooth. Stir in 1 cup of granulated sugar.
Whisk in 2 large eggs one at a time, then stir in 0.5 teaspoon vanilla extract.
In a separate bowl, whisk together 0.5 cup all-purpose flour, 0.25 teaspoon salt, and 0.25 cup unsweetened cocoa powder.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Do not overmix.
In a medium bowl, beat together 8 ounces softened cream cheese and 0.33 cup granulated sugar until smooth. Stir in 1 large egg yolk and 0.5 teaspoon vanilla extract.
Spread half of the brownie batter into the prepared baking pan. Dollop spoonfuls of the cheesecake mixture over the brownie batter.
Gently spread the remaining brownie batter over the cheesecake mixture, then swirl the two batters together with a knife or toothpick.
Bake for 30-35 minutes, or until a toothpick inserted into the center comes out with moist crumbs attached. Do not overbake.
Let cool completely in the pan on a wire rack before cutting into squares.
Notes
For an even richer flavor, use a good quality dark chocolate.